Decomposition of the maximal nilpotent subalgebra of the simple Lie algebra

Question

Let $\mathfrak{g}$ - simple Lie algebra, $\mathfrak{g}=\mathfrak{n}\oplus \mathfrak{h} \oplus \mathfrak{n}_-$ - Cartan decomposition, $\Phi$ - root system, $\Phi^+$ - positive roots w.r.t. $\mathfrak{n}$. Let $\beta$ - maximal root of $\Phi$. If $\alpha \in \Phi$, then $e_{\alpha}$ - cor. root vector. $\Phi^+_1=\{ \alpha,\alpha' | \alpha+\alpha'=\beta; \alpha, \alpha' \in \Phi^+ \}\cup \{ \beta \} $, $\mathfrak{s}=\langle e_{\alpha} | \alpha \in \Phi^+_1 \rangle$, $\mathfrak{n}'=\langle e_{\alpha} | \alpha \in \Phi^+ \setminus \Phi_1^+ \rangle$. If it exists, I want to know article, where prove that $\mathfrak{n}=\mathfrak{n}_1 \rtimes \mathfrak{s}$. I can prove it, but I want to сite.